{"id":"928a6f82-a4a7-421d-8af1-808a2dbaab09","arxiv_id":"2411.14097","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves, for a restricted family of automorphisms, that the Mordell-Weil group of an elliptic curve over the fixed field has infinite rank.","lead":"A number theory preprint claims a proof of Larsen's conjecture on elliptic curve ranks for a special class of Galois automorphisms. The argument combines Heegner points with a parity trick on class numbers, but it contains unproved assumptions about the Galois families it uses.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.4 leans on a citation that may not cover arbitrary elliptic curves; without it the infinite-rank conclusion is unsupported, while the reader's Σ_j-existence worry is repairable.","rationale":"I read the paper as trying to prove a special case of Larsen's conjecture: for a finitely generated G whose generators all lie in one family Σ_j, the fixed field contains infinitely many independent Heegner points. The main structural steps for this strategy are: (a) Σ_j is nonempty/infinite, (b) each prescribed involution fixes a Heegner point on the modular curve, and (c) the images of these points under a modular parametrization are independent. Step (b) is essentially correct: for a finite set of odd cardinality, an involution has a fixed point. Step (a), which the reader identifies as the weakest assumption, can in fact be justified. Since H_p/Q is Galois and ramified only at p, the family {H_p} is strongly linearly disjoint: any subfield of a finite compositum of the H_p's ramifies only among the corresponding primes, so it cannot meet a new H_q nontrivially. Consequently any choice of involutions ψ_jp defines a compatible system on the compositum and extends to a global automorphism, and multiplying by the nontrivial absolute Galois group of the compositum makes the family infinite. Thus I do not think the paper is vacuously true for the reason the reader states. Step (c) is where the argument is least secure. The manuscript's Theorem 3.4 is a direct quotation of an external theorem, but the quotation as printed has an internal inconsistency (E(Q) vs. E(Qbar)) and the cited paper's title suggests a scope restriction to CM elliptic curves. Since the entire infinite-rank conclusion rests on the independence of the Heegner points, a misquoted or misapplied independence theorem is the single most load-bearing concern. The right check is to compare the manuscript's Theorem 3.4 verbatim with [Sah13] Theorem 1.1 and, if necessary, with [RS07]; until that is done, the present version should not be accepted as proving the claimed theorem for arbitrary E.","tokens_in":6005,"tokens_out":31541,"duration_ms":338977,"concrete_test":"Locate Şahinoğlu, Proc. AMS 141 (2013), Theorem 1.1 and check: (1) Does it assume the elliptic curve E has complex multiplication, or otherwise restrict E beyond being a modular elliptic curve of conductor N? (2) Does its independence conclusion hold in E(Qbar)/tors, with the class-number lower bound depending only on E and Φ_E, exactly as quoted? If either answer is no, check whether Rosen–Silverman [RS07] provides the missing general case and verify its hypotheses (e.g., any additional conditions on discriminants or class numbers) against the fields k_p constructed in Lemma 3.2. If no applicable theorem covers arbitrary E/Q, Theorem 3.5's infinite-rank conclusion is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central proof of Theorem 3.5 depends on Theorem 3.4, which the manuscript justifies by 'See Theorem 1.1 in [Sah13]'. The quotation itself is suspect: it states independence in E(Q)/E_tors, although the Heegner points P_i are generally not defined over Q, and the application immediately changes the field to E(∏ H_{p_i}). More seriously, the cited paper's title indicates it concerns Heegner points on CM elliptic curves, i.e., it may assume the elliptic curve E has complex multiplication. Theorem 3.4, however, is asserted for an arbitrary elliptic curve E over Q, and Theorem 3.5 uses it for every modular elliptic curve. If [Sah13] Theorem 1.1 has a CM hypothesis, or imposes a different class-number condition, then the independence of the infinite family {P_p} is not established for non-CM E, and the rank conclusion in Theorem 3.5 collapses. The reader's weakest-assumption about nonemptiness of Σ_j is less damning: because each H_p/Q is Galois and ramified only at p, any finite sub-compositum of the H_p has ramification supported on a finite set of primes, so it is linearly disjoint from any H_q with q outside that set; hence arbitrary prescribed involutions ψ_jp glue to a global automorphism of the compositum, extend to Gal(Qbar/Q), and Σ_j is genuinely nonempty (indeed infinite). The unverified independence theorem is therefore the more serious defect.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to prove Larsen's conjecture on the infinite rank of E(Qbar^G) for finitely generated subgroups G of Gal(Qbar/Q) when each generator belongs to one of certain infinite families Sigma_j of Galois automorphisms. The strategy is to take infinitely many imaginary quadratic fields k_p with odd class number satisfying the Heegner hypothesis for the conductor N, attach to each its Hilbert class field H_p, and use Heegner points on X0(N) that are fixed by a chosen involution on each H_p. The authors define Sigma_j as the set of global automorphisms whose restriction to every H_p is the j-th involution of Gal(H_p/Q), assert that Sigma_j is an infinite family, and then use a cited independence theorem for Heegner points to conclude that the image of the fixed Heegner points under a modular parametrization gives an infinite independent set in E(H_E^G), hence infinite rank. The central claim is therefore a partial result toward Larsen's conjecture for a special class of finitely generated groups.","tokens_in":6303,"tokens_out":17702,"duration_ms":158320,"significance":"If the proof were correct, the paper would provide a new and interesting partial result toward Larsen's conjecture, going beyond the cyclic case previously treated by Im and others. The idea of using a 'broad' family of Hilbert class fields instead of a single deep ring class field is attractive and could be a useful technique. The paper ships no code and no machine-checked proofs, but the strategy is conceptually clear. However, the significance is conditional on resolving two load-bearing issues: the nonemptiness of the families Sigma_j is asserted without proof, and the independence theorem cited as Theorem 3.4 is misstated and is drawn from a source whose title indicates a CM hypothesis, raising serious doubt about its applicability to arbitrary elliptic curves. These gaps currently prevent the paper from establishing its main theorem.","major_comments":[{"comment":"Theorem 3.4 is not correctly stated and its proof by citation is not sufficient. The points P_i = Phi_E(y_i) are defined over the Hilbert class fields H_{k_i}, not over Q (Theorem 2.5), so the phrase 'independent in E(Q)/E_tors' is meaningless unless all P_i happen to lie in E(Q), which is false in general. The application in the proof of Theorem 3.5 uses independence in E(prod H_{p_i})/tors, so the theorem statement should be corrected accordingly. More seriously, the cited result, Theorem 1.1 in [Sah13], is titled 'On the independence of Heegner points on CM elliptic curves associated to distinct quadratic imaginary fields'; if that theorem indeed assumes E has complex multiplication, then it cannot be used to establish Theorem 3.4 for an arbitrary elliptic curve over Q, and the infinite-rank conclusion in Theorem 3.5 for non-CM E is unsupported. The authors must either quote the exact theorem and verify that its hypotheses hold for all E, or supply a correct reference or proof that covers arbitrary modular elliptic curves. This issue is load-bearing because the proof of Theorem 3.5 depends entirely on the independence of the family {P_p}.","section":"Theorem 3.4"},{"comment":"The nonemptiness of Sigma_j is asserted without proof. After defining Sigma_j := {sigma in Gal(Qbar/Q) | sigma|_{H_p} = psi_{jp} for all p in A_N}, the manuscript states 'Sigma_j is an infinite family' but gives no argument that such a global automorphism exists. If no such sigma exists, then Theorem 3.5 and Corollary 3.7 are vacuous. The existence can be proved by noting that the Hilbert class fields H_p are linearly disjoint over Q (their discriminants are supported on distinct primes), so any choice of automorphisms psi_{jp} on each H_p is compatible on finite composita, and then a global automorphism exists by compactness of Gal(Qbar/Q). This argument should be included. Since the hypothesis of Theorem 3.5 requires sigma_i in Sigma_j, this is a load-bearing missing step.","section":"Section 3, definition of Sigma_j"}],"minor_comments":[{"comment":"The proof of Corollary 2.7(1) is hard to follow as written, partly because the text 'sqrt(-d) or (tau sigma)|_k = 1' should read 'sqrt(-d), so (tau sigma)|_k = 1'. The computation is correct after this typo is fixed, but the presentation should be clarified.","section":"Corollary 2.7(1)"},{"comment":"In the proof of Theorem 3.5, the phrase 'sigma_i(y_jp) = psi_{jp}(y_jp) = y_jp, or y_jp in H_p^G' should use 'hence' instead of 'or' to avoid confusion.","section":"Theorem 3.5 proof"},{"comment":"The notation 'H_E^G < Qbar^G' is inappropriate because the fixed fields are fields, not groups; use 'subset' or 'subfield' notation.","section":"Corollary 3.7"},{"comment":"Theorem 3.9 is stated too vaguely: the 'infinite families Sigma' are not defined, and the proof merely says 'one can do the same as in the Proof of Theorem 3.5'. This theorem should either be made precise or removed.","section":"Theorem 3.9"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is Theorem 3.4. If Theorem 1.1 of [Sah13] indeed has a CM hypothesis, the manuscript's main theorem is unsupported for general elliptic curves and the paper should be rejected. I recommend asking the authors to verify the exact statement of [Sah13] and to provide a valid independence result for all elliptic curves, or to restrict the main theorem to the case where such a result is available. The nonemptiness of Sigma_j is easily repairable and should not by itself block publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the kernel is real. The observation that an odd class number forces any involution of the Hilbert class field to fix a Heegner point is new and neat, and the resulting theorem for finitely generated groups whose generators all act by the same prescribed involution on each H_p is not in the cited literature. That is honest progress for a narrow multi-generator family, not a breakthrough.\n\nThe load-bearing soft spot is Theorem 3.4. It is stated for an arbitrary elliptic curve over Q and cites Theorem 1.1 of [Sah13], but the title of [Sah13] says \"on CM elliptic curves.\" If that theorem has a CM hypothesis, it does not cover non-CM E, and the proof of Theorem 3.5 collapses for those curves. The statement also claims independence in E(Q)/E_tors, while the Heegner points P_i are generally not in E(Q); the application silently switches to E(∏ H_{p_i})/tors. That may be a typo, but combined with the CM issue the central independence claim is unsupported. If [RS07] actually contains the needed result for all E, the paper could be repaired by a different citation; a referee would need to check that.\n\nThe other gaps are minor. The assertion that Σ_j is nonempty is unproved, but a one-paragraph linear-disjointness argument (each H_p ramifies only at p) supplies the global automorphism, so that worry is overblown. The proof of Corollary 2.7(1) is garbled, though the statement is a standard dihedral fact. Lemma 3.2 and the Dirichlet construction look correct.\n\nNet: the idea is worth engaging, but the paper as written does not support the theorem for arbitrary elliptic curves. I would not desk-reject it; it deserves a referee who knows the Heegner-independence literature and can adjudicate the [Sah13] vs [RS07] question. But I would not cite it yet.","headline":"Neat fixed-point trick for Heegner points under involutions, but the independence theorem it leans on is mis-cited (CM vs arbitrary E), leaving the main theorem unsupported as written.","tokens_in":6822,"tokens_out":12481,"would_cite":false,"duration_ms":113593,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11G18","11R29","11R37"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves Larsen's conjecture for elliptic curves over Q when the generators of a finitely generated Galois group come from one of the paper's infinite families of Galois automorphisms.","keywords":["elliptic curves","Larsen's conjecture","Mordell-Weil rank","Heegner points","Hilbert class fields","Galois fixed fields","modular parametrization"],"falsifier":"Find two primes $p,q \\in A_N$ for which the chosen involutions $\\psi_{jp}$ and $\\psi_{jq}$ cannot be realized simultaneously by any automorphism of the compositum $H_pH_q$; then $\\Sigma_j$ is empty and Theorem 3.5 has no instances. A concrete route is to compute the Galois group of $H_pH_q/\\mathbb{Q}$ and check whether the local prescriptions agree on the intersection of the two fields.","tokens_in":5757,"feed_emoji":"📈","tokens_out":14451,"duration_ms":127322,"temperature":0.7,"pith_summary":"Larsen's conjecture predicts that for every elliptic curve $E$ over $\\mathbb{Q}$ and every finitely generated subgroup $G$ of $\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})$, the group of rational points on $E$ over the fixed field $\\overline{\\mathbb{Q}}^G$ has infinite rank. The paper proves the conjecture for the case in which each generator of $G$ belongs to one fixed infinite family $\\Sigma_j$ of Galois automorphisms, families built by prescribing an involution on each Hilbert class field in an infinite family attached to imaginary quadratic fields of odd class number. The strategy produces an infinite set of independent Heegner points that are individually fixed by every generator of $G$, so their images on $E$ form an infinite independent set in $E(H_E^G)$. Because $H_E^G$ sits inside $\\overline{\\mathbb{Q}}^G$, the same conclusion transfers to $E(\\overline{\\mathbb{Q}}^G)$, giving Larsen's conjecture for these groups.","feed_headline":"Elliptic curve ranks proven infinite over broad Galois fixed fields","feed_subtitle":"New Heegner-point argument forces infinitely many independent rational points fixed by the group.","key_machinery":"The central mechanism is a family of Heegner points living in 'broad' Hilbert class fields $H_p$ rather than in a tower of 'deep' ring class fields. For each prime $p$ in the infinite set $A_N$, the imaginary quadratic field $k_p = \\mathbb{Q}(\\sqrt{-p})$ has odd class number $h_p$ and satisfies the Heegner hypothesis for the conductor $N$, so $\\operatorname{Gal}(H_p/\\mathbb{Q})$ contains exactly $h_p$ involutions. Fixing one index $j$ selects an involution $\\psi_{jp}$ on every $H_p$, and the oddness of $h_p$ forces any such involution to fix at least one of the $h_p$ Heegner points $y_{jp}$ on $X_0(N)$ attached to $k_p$. The modular parametrization $\\Phi_E : X_0(N) \\to E$ sends each $y_{jp}$ to a Heegner point $P_p$ on $E$ defined over $H_p$, and an independence theorem for Heegner points attached to distinct imaginary quadratic fields ensures that, once the odd class numbers exceed a constant $C(E,\\Phi_E)$, the points $P_p$ are non-torsion and independent. Since every generator $\\sigma_i$ of $G$ restricts to $\\psi_{jp}$ on $H_p$, all $y_{jp}$ are fixed by $G$, hence all $P_p$ lie in $E(H_E^G)$, forcing infinite rank.","core_discovery":"The central claim is Theorem 3.5. For an elliptic curve $E$ over $\\mathbb{Q}$ of conductor $N$, if $G = \\langle\\sigma_1,\\dots,\\sigma_n\\rangle$ is a finitely generated subgroup of $\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})$ whose generators all lie in one family $\\Sigma_j$, then the rank of $E(H_E^G)$ is infinite; here $H_E$ is the compositum of the Hilbert class fields $H_p$ attached to the infinite family of imaginary quadratic fields $\\mathbb{Q}(\\sqrt{-p})$ with $p \\in A_N$. Since $H_E^G$ is a subfield of $\\overline{\\mathbb{Q}}^G$, the same infinite-rank conclusion holds for $E(\\overline{\\mathbb{Q}}^G)$, which is exactly Larsen's conjecture for these groups. The paper also records a general version (Theorem 3.9) for any infinite family of imaginary quadratic fields satisfying the Heegner hypothesis whose odd class numbers exceed a constant depending on $E$ and its modular parametrization, and it exhibits a nested chain of $G$-fixed subfields over each of which the Mordell-Weil rank is infinite.","pith_inferences":["The pivotal unverified step is the lifting of the local involutions $\\psi_{jp}$ to a single global automorphism of $\\overline{\\mathbb{Q}}$; if this compatibility can be proved, the families $\\Sigma_j$ become concrete rather than conditional, and the main theorem applies to actual groups.","The fixed-index restriction on $j$ appears to be an artifact of the proof: the argument only needs the generators to preserve the chosen Heegner points $y_{jp}$, so generators drawn from different families might work whenever the corresponding global automorphisms exist.","The same broad-field construction should transfer to other modular settings: any quotient of a modular curve with an independence theorem for its Heegner points would yield infinite rank over the corresponding fixed fields, so the method is not inherently limited to elliptic curves."],"forward_implications":["For any elliptic curve $E/\\mathbb{Q}$ and any finitely generated $G$ whose generators lie in a single family $\\Sigma_j$, the Mordell-Weil group $E(\\overline{\\mathbb{Q}}^G)$ has infinite rank, so Larsen's conjecture holds for these groups.","The rank is infinite already over the much smaller field $H_E^G$, the $G$-fixed subfield of the compositum of the Hilbert class fields.","For every infinite subfamily $A' \\subseteq A_N$, the rank of $E\\left(\\left(\\prod_{p\\in A'} H_p\\right)^G\\right)$ is infinite, so deleting finitely many primes does not destroy the result.","Removing the first few primes from $A_N$ produces a nested chain of $G$-fixed subfields inside $H_E^G$, and each of these subfields still carries a Mordell-Weil group of infinite rank."],"supporting_citations":[{"why":"States the conjecture that $E(\\overline{\\mathbb{Q}}^G)$ has infinite rank for finitely generated $G$; this is the statement the paper sets out to prove.","marker":"[Lar03]"},{"why":"Proves the one-generator case and supplies the Heegner-point strategy extended here to arbitrary finite generation.","marker":"[Im07]"},{"why":"Supplies the odd-class-number criterion, ring class field facts, and the class-number growth estimate used to build the infinite family $A_N$.","marker":"[Cox89]"},{"why":"Provides the characterization of Heegner points on $X_0(N)$ and the modular parametrization theorem used to move them to $E$.","marker":"[Dar04]"},{"why":"Gives the full conjugacy set of a Heegner point and the involution structure of $\\operatorname{Gal}(H_p/\\mathbb{Q})$ used to force a fixed Heegner point.","marker":"[RS07]"},{"why":"Supplies the modular parametrization $X_0(N)\\to E$ for elliptic curves over $\\mathbb{Q}$.","marker":"[DS05]"},{"why":"Supplies the independence theorem for Heegner points attached to distinct imaginary quadratic fields, which makes the produced points non-torsion and independent.","marker":"[S ¸ah13]"}],"fun_headline_variants":["Larsen's conjecture proven for elliptic curves over Galois fixed fields","Infinite rank for elliptic curves under Galois fixed fields","Heegner points force infinite rank on elliptic curves over Galois fixed fields","Infinite ranks over Galois fixed fields: Larsen's conjecture proven","Larsen's conjecture resolved: infinite ranks over Galois fixed fields"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire construction depends on the existence of a single automorphism $\\sigma \\in \\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})$ whose restriction to every Hilbert class field $H_p$ is the chosen involution $\\psi_{jp}$; the paper asserts that the family $\\Sigma_j$ is infinite without proving that these local involutions are compatible, so if no such $\\sigma$ exists the main theorem is vacuous.","fun_headline_variants_meta":{"raw":{"variants":["Larsen's conjecture proven for elliptic curves over Galois fixed fields","Infinite rank for elliptic curves under Galois fixed fields","Heegner points force infinite rank on elliptic curves over Galois fixed fields","Infinite ranks over Galois fixed fields: Larsen's conjecture proven","Larsen's conjecture resolved: infinite ranks over Galois fixed fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000596,"raw_usage":{"total_tokens":2777,"prompt_tokens":919,"completion_tokens":1858,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":1765}},"tokens_in":535,"tokens_out":1858,"duration_ms":13604,"temperature":1.0,"reasoning_tokens":1765,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:34:02.893111+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find two primes $p,q \\in A_N$ for which the chosen involutions $\\psi_{jp}$ and $\\psi_{jq}$ cannot be realized simultaneously by any automorphism of the compositum $H_pH_q$; then $\\Sigma_j$ is empty and Theorem 3.5 has no instances. A concrete route is to compute the Galois group of $H_pH_q/\\mathbb{Q}$ and check whether the local prescriptions agree on the intersection of the two fields.","supporting_citations":[],"review_version":1}